Sunday, April 23, 2017

Newton and Leibniz

No one got the joke. The office  has a lot of nerds. Our boss earned his master's in anthropology.  My wife said, "The human brain receives 10 million bytes of data per minute and focuses on 32: cookies."


I discovered the Leibniz either at the Wheatsville Co-op or my neighborhood Whole Foods.  I had to goto four grocery stores to find the Fig Newtons.

Previously on Necessary Facts

Monday, April 10, 2017

Grigory Perelman's "Perfect Rigor" by Masha Gessen

Technical errors in common mathematics and common English from a writer who claims to an early love for mathematics, and who is professionally literate in two difficult languages (Russian and English) leave the book suspect. Masha Gressen achieved fame for her success as a journalist. Her specialty is the politics of gender. A strong advocate in Russia for gay rights, she fled (back) to the United States shortly after a personal meeting with Vladimir Putin.  That only makes it harder to understand how she could have put her name on such a weak work as this book. 
Perfect Rigor: A Genius and
the Mathematical Breakthrough
of the Century

by Masha Gessen,
Houghton Mifflin Harcourt, 2009.
Page 135: “Indeed, it is easy to see that on this [spherical] surface, any two straight lines-a straight line being the extension of a segment that connects two points in the shortest possible way-will cross. All straight lines on the apple, or on the Earth, are “great circles” with the centers at the center of the sphere."
Page 135: "Not all of us travel so far all the time, but in the imagination - the very place where mathematics resides - the shortest distance between two points is the trajectory described by an airplane, which generally lies along a geodesic, even if we have never hears the word. These straight lines do not go on forever, but, being circles, inevitably close in on themselves. And, of course, they cross, any two of them."
cal state long beach rodrig geog 140 parallel.jpg
How the parallels of latitude do not contradict that is not clear to me. Indeed, it is possible to draw parallels all over the surface of a sphere. They are concentric circles. They can be any place and of any convenient size. You can start by taking any line on a flat map from Hometown to Smallville and drawing a track parallel to it. Drawing the path on a sphere reveals them to be sections of curves with common origins.  It is easy for me to accept that I am missing some critical piece of common information about spherical geometry. But in that case, this book then has another problem: too little is explained. 
Riemann sphere maps to a plane (Encyclopedia Britannica)
Homeomorphic parallel lines are obvious by inspection.
[13 April 2017 - Still spending time with this ...  After I posted the article, I continued googling and found proofs that the shortest distance between two points on the surface of a sphere is a segment of a great circle.  See Wikipedia here and Wolfram Mathworld here. Driving back and forth to work, I visualized two points on an "arctic circle" and then imagined connecting them with an arc segment of a great circle. Rational proofs are nice, especially in mathematics, but I would like to try it with string and a soccer ball.]

Page 143: "Think about a simple function of the sort you studied in high school. Say, 1/x. A graph of this function would look like a smooth line until it got to the point where x=0. Then things would get crazy because you cannot divide by zero. The line of your graph would suddenly soar toward eternity. This is called a singularity."

First, while colloquial writing is fine for common communication, the expression 1/x is not a function. The proper statement - and it is a statement - is of the form f(x) = 1/x or y = 1/x.
Furthermore, the line would still be "smooth" i.e., continuous all along its path. It would not "suddenly" soar; and you could change the apparent "soar" just by changing the scale of the graph. And, in any case, while half of the lines would rise up or down - and down is diving not soaring - the other halves would creep ever closer to the horizontal positive or negative. Finally, the distinction between eternity and infinity might matter most only to philosophers and theologians, but the difference exist nonetheless.
Graph of f(x) = 1/x
zonalandeducation.com 
And no one else seemed to have my difficulties.  I found glowing reviews for this book from the New York Times, the American Mathematical Society, and the Mathematical Association of America. 
SUNDAY BOOOK REVIEWhttp://www.nytimes.com/2009/12/13/books/review/Hoffman-t.htmlGrigori Perelman’s Beautiful MindBy Jascha Hoffman Dec. 10, 2009
NOTICES OF THE AMS VOLUME 58, NUMBER 1http://www.ams.org/notices/201101/rtx110100056p.pdfPerfect Rigor: A Genius and the Mathematical Breakthrough of the CenturyReviewed by Donal O’Shea
MAA REVIEWShttp://www.maa.org/press/maa-reviews/perfect-rigor-a-genius-the-mathematical-breakthrough-of-the-century[Reviewed by Darren Glass on 01/17/2010]


However, despite all of that, or perhaps because of it, I was motivated to search for “Poincaré’s Conjecture” on YouTube and I found several explanations. The best was by Rendell Heyman but each of them helped in some way.  I first found Heyman in an archived panel discussion of the Poincaré  Conjecture from the World Science Festival. Heyman offers several YouTube channels dedicated to explaining mathematics and some technology and science. His website is here

And the UT Libraries shelve several books on Poincaré s Conjecture.  So, I have some reading to do. 
  • Ricci flow and the Poincaré conjecture  by John W. Morgan
  • The Poincaré conjecture: in search of the shape of the universe by Donal O'Shea.
  • Poincaré's prize : the hundred-year quest to solve one of math's greatest puzzles by George Szpiro
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Monday, April 3, 2017

Mapping it Out: An Alternative Atlas of Contemporary Cartographics

This is a quirky book that challenges the reader to accept unusual presentations of unusual information. Among the artists are Bruce Sterling, Yoko Ono, and Tim Berners-Lee. Mapping it Out: An Alternative Atlas of Contemporary Cartographics edited by Hans Ulrich Obrist with an introduction by Tom McCarthy, Thames & Hudson, 2014.  
The Size of Your Senate Vote by artist James Croak pages 66-67.
Edward Tufte’s The Visual Display of Quantitative Information is the classic work in this genre. His website is here. Unlike that book, this one speaks for and to cartographers.  It breaks many rules that bibliophiles accept implicitly. For example all of the front matter – copyright, publication – is in the back.  The table of contents (in the front) is in the format of a map key. 
 
Africa by Kai Krause pages 44-45.
The five chapters across 240 pages are: Redrawn Territories; Charting Human Life; Scientia Naturalis; Invented Worlds; and The Unmappable. Presentations consist of two facing pages (occasionally one) with a map on one (usually the right, odd-numbered) and a key to the author and the work on the other. 
Mind Map of Western Philosophy on the Coppelia website here.
The maps of "Product Space" by César Hidalgo (page 66-67) and of diseases by Albert-László Barabási (pages 142-143) reminded me of the Mind Map of Western Philosophy on the Coppelia website here.  Coppelia offers solutions in machine learning and analytics. That image is from their blog for 13 June 2012 by Simon Raper, posted in Data. It was reproduced on the Coppelia blog two years later for a different discussion.
 
Mycoplasma Mycoides JCVI-synth 1.0
by J. Craig Venter, pages 146-147.
See also a road map of viruses by George Church page 128. 
Some of the maps are startlingly mundane, such as the black-and-white aerial photo of central Oslo (pages 188-189), or the drawings of the northern polar region by earth scientist Laurence C. Smith.  On the other hand, Tim Berners-Lee’s projection of what cyberspace looks like to him reminded me of Bilbo Baggins’s rendering of Middle Earth – and perhaps that is deeply appropriate.
 
Transactions define regions by Carlo Ratti page 121.
These 3-D stacks show telecommunication units.
Most people in most places call their neighbors 
but London reaches out. 
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Sunday, April 2, 2017

Elisha S. Loomis and the Pythagorean Proposition

The fact that over 300 proofs of the Pythagorean Theorem are known and an infinite number are possible is in and of the nature of the universe.  When something is true, many other truths support it; and a truth leads to many others.  Moreover, no truth can be supported by a fallacy; and a truth cannot lead to a fallacy. 

The Pythagorean Proposition: Its Proofs Analyzed and Classified and Bibliography of Sources for Four Kinds of Proofs by Elisha Scott Loomis, Ph. D., LL.B., Masters and Wardens Association of the 22nd Masonic District of the Most Worshipful Grand Lodge of Free and Accepted Masons of Ohio, Cleveland, Ohio, 1927.  National Council of Teachers of Mathematics, Washington, D.C. 20036, 1940, 1968; 2nd printing 1972.

Prof. Elisha S. Loomis divided proofs of the Pythagorean Theorem into four classes: algebraic, geometric, quaternionic, and dynamic. 

Algebraic proofs show that for two numbers, each of them multiples of other numbers by themselves (squared), there exists a third number, also the product of some number by itself (a square) that is the sum of those.  That is, for some integers A and B there is a C such that C*C equals A*A + B*B.  Geometric proofs are achieved by constructions, folding, cutting, and superimpositions. Dynamic proofs come from the mathematics of engineering mechanics, from vector arithmetic, in which magnitude and direction together define a force. 

For quaternions, I can only quote Wikipedia: “quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative."  In other words A * B ≠B * A.  The only explanation I know – perhaps not actually from quaternions – is that there are ways to flip a book (transformation with rotation) so that when you reverse the steps, the book is in a different orientation than when you started.

But I could follow the concepts supporting the algebraic and geometric proofs; and they are the foundation for the vector arithmetic.

The simplest algebraic proofs are attributed to Pythagoras and Euclid, but I would like to find them in the original because, in fact, those ancient savants did not know algebra. Nonetheless, citing Loomis, those proofs were explored and generalized by Artemas Martin (1891) and Leonard E. Dickson (1894).

Start with any odd number. Square it and subtract 1; then divide by 2; this is the second number. For the third, take the original, add 1 to it, and square the sum. That third number equals the sum of the squares of the other two.

X = 2n + 1  is any odd number; square this.
((X^2) – 1) / 2 becomes the second number; square this.
(X+1)^2 is the third number in the triplet.

The algebraic proof is difficult to lay out with a word processor but is easy enough with paper and pencil.

The simplest geometric proof is achieved by starting with a right triangle. Drop a perpendicular from the right angle. The three triangles are similar. The ratios and proportions of their respective sides can be stated algebraically, and then be manipulated to show that the square of the original hypotenuse is equal to the sum of the squares of the other two sides.

The other 368 proofs are left for the reader to enjoy.

Elisha Scott Loomis held a doctorate in mathematics from Wooster College in Ohio, earned a law degree and was admitted to the Ohio bar, worked as the town engineer in Beria, Ohio, and, after some other appointments and assignments became the chairman of the mathematics department at West High School in Cleveland, Ohio. 

The first edition of this book includes a roster of the 22nd Masonic District of Ohio with their officers, including the Past Grand Masters, and the Masters and Wardens Association, about 150 names in all. I recognized many of the neighborhoods:  Brooklyn, Denison, Collingwood, Windemere, Lakewood, Euclid, Sheffield. Other lodges had honorific names: Laurel, Pythagoras, Warren G. Harding, North Star, Halcyon.

The second edition had three printings, 1940, 1968, and 1972 by the National Council of Teachers of Mathematics. Their office is listed in Washington DC, but the books were printed in Ann Arbor. The dedications to the FAM lodges were removed in the reprinting.

You can find a biography of Elisha Scott Loomis in Wikipedia.  The same information can be found in the front matter of the books.  Other biographies have been archived online.

“While he was living in Berea, Loomis began to study civil engineering and became the village engineer. He also served a three-year term as president of the Berea board of education. He continued to study while he taught, and this led to an A. M. degree from Wooster (Ohio) University in 1886 and a Ph.D. in metaphysics and social science from that same institution in 1888. The subject of his doctoral thesis was "Theism the Result of Completed Investigation."

In 1895 Professor Loomis left Baldwin University to become head of the mathematics department at Cleveland West High School, where he remained until his retirement in 1923. In 1900 he earned the LL.B. degree from the Cleveland Law School and was admitted to the Ohio state bar. He served a 3-year term as president of the High School Teachers¼ Mathematics Club of Cleveland, and he was active in the Masonic lodge.” – Kullman ("Elisha Scott Loomis," The National Cyclopaedia of American Biography, vol. 15, New York, James T. White & Company, 1914, p. 186. Article by David E. Kullman, Miami University.)


Amazon Ripoff 
https://www.amazon.com/gp/offer-
listing/0873530365/
ref=dp_olp_used_mbc?
ie=UTF8&condition=used
ELISHA SCOTT LOOMIS, educator, author and man of affairs of Lakewood, was born on a farm in Wadsworth Township, Medina County, this state, September 18, 1852, son of the late Charles Wilson and Sarah (Oberholtzer) Loomis.    He was superintendent of schools at Shreve, Ohio, from 1876 to 1879; principal of Burbank Academy at Burbank, Ohio, from 1880 to 1881; principal of Richfield Central High School, Summit County, Ohio, from 1881 to 1885; professor of mathematics in Baldwin Wallace University from 1885 to 1895, and from 1895 up to the close of 1923 was head of the department of mathematics in West High School of Cleveland. At the last mentioned date he retired on school pension, having reached the limit established by Ohio law which provides that no teacher of the public schools shall hold position after having reached the age of seventy years. He taught his first school, beginning in April, 1873, and completed his last term of teaching in June, 1923, thus rounding out a full half century of successful teaching, and but for the intervention of the Ohio school law limiting the age of teachers for school teaching he would have continued his school work for an indefinite period, should he have so desired, for his physical and mental faculties are unimpaired, and "his spirit is willing."
http://www.onlinebiographies.info/oh/cuya/loomis-es.htm

A caveat regarding Loomis’s books for sale on Amazon: Despite the exorbitant price ($150), this book is clearly some kind of photocopy, rebound with a GBC or similar coil. It is a cheap, modern violation. Other sellers apparently are offering the real thing, the 1940 edition or lawful reprints, at reasonable prices.

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